omogeneous nucleation in pure liquids has been H treated by different postulates. In 1926 Volmer (27) considered nucleation to follow thermotwo
dynamic equilibrium states to an unstable condition. Others have considered it to be the result of a statistical density fluctuation. Various theories are discussed in references (2-6,8). Common to the two approaches is the requirement for a superheated liquid. For a bubble to grow, the superheat must be great enough to overcome surface tension effects tending to collapse very small bubbles. Figure 1 shows a conventional P-V-T diagram for a single pure substance. The lines AB and FG represent isotherms in the liquid and vapor phases. Isothermal expansion of the liquid at A ordinarily results in boiling at PB; however, if the liquid is of good purity and the container material is very clean and does not react with the liquid, it is possible to proceed as a liquid down to point C which may actually be a negative (tension) preasure. Brig@ (6) subjected water at room temperature to negative pressures of nearly 270 atm. by spinning a “Scrupulously clean” open-ended tube of pure water about a vertical axis perpendicular to tube axis. The centrifugal force required to break the liquid determines the pressure at fracture. Figure 2 shows some of his results. The portions BC and EF in Figure 1 represent metastable state-uperheated liquid and subcooled vapor; the portion CDE is completely unstable and cannot exist. The locus of limiting points, CR,represents what Gibbs (77) called the “Limit of Essential Instability,” (ap/b)r = 0. Approximate expressions for the line ABCDEFG, and hence, CR, are given by the van der Waal equation and others (75). A liquid at H, heated at constant pressure may theoretically be heated to point R before it becomes unstable forming a vapor bubble. The data of Wismer (24) I are compared with this theory in Figure 3. These data were obtained under very carefully controlled conditions of cleanliness. Actually, boiling from a solid surface occurs at superheats of a few degrees at atmos-
pheric pressure, where this equilibrium theory predicts 90” to 100” F. Obviously then, this theory does not describe the nucleation process at a solid-liquid interface in commercial equipment. Thermodynamic Equilibrium at a Curved Interface
The conditions of equilibrium at a curved interface yield relations useful in predicting nucleation conditions in heat transfer systems which, of course, are not in equilibrium. The following equilibrium conditions must hold (76) for a liquid in contact with its own vapor: -Uniform temperature throughout both phases -Both phases must have equal chemical potentials (for a pure substance, this is the Gibbs free energy) -For curved interfaces
where rl and rt are principal radii of curvature. For a spherical surface, r1 = rt = 7, or
p.-p=
2U
This latter relation follows directly from a force balance of a static spherical bubble. Consider a capillary tube piercing the liquid-vapor surface in a container with a pure substance at a uniform temperature, Figure 4. The liquid level in the tube will rise a distance, y. The pressure at the flat interface is the usual pastcorresponding to the temperature. Because of the hydrostatic head, the pressures on each side of the curved interface in the tube are
pw
=
pa.*
- P@ - PIXY
Ob) Clearly the pressures at the curved interface are both below peat; hence, both the liquid and the vapor are superheated to the normal saturation condition-the vapor being only slightly superheated since po is very small except near the critical point. These conditions, shown in Figure 5, approximate the equilibrium conditions at the interface of a vapor bubble in a liquid (8). fir
= Peat
WARREN M. ROHSENOW
nucleation with boiling heat transfer nucleation symposium 40
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
(24
Even though we don't thoroughly understand the phe-
nomenon of boillng, we can come to grips with it by idealizing the conditions for nucleation to include those
cases of practical interest to the profit minded designer
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The condition alow the normal saturation curve is given by the Clapeyron equation:
dP hY& (3) dT R,P which here includes the perfect gas law. If the vapor is assumed to be at the normal saturation state, a good approximationas shown in Figure 5, Equation 3 may be integrated between p and fn the conditions acma a bubble interface: T - T . . * = R-InP 1+(4) h, or ifp >> 2 u/r, 2 R.T% T - T,t = (5) hYgr This represmts an approximate e x p r 4 o n for the superheat required for equsbrium of a bubble of radius r. Nuclei of radius greater than r from Equation 5 should become 'bubbles and grow; those of smaller radius should collapse. Some experiments have shown that at a heated surface in watex at atmmph&ic. pressure, boiling begins at about 30' F. above saturation temperature. For this condition, Equation 5 predicts an equilibrium bubble radius of lo-' inch. This is about 10,000 times larger than the maximum cavity size expected from molecular fluctuations (70). Volmer (27) estimated a cavity formation rate of size lo-' inch to be approximately one per cubic inch per hour. From this, we readily conclude that free vapor nuclei arising from molecular fluctuations are not important as nucleation cavities. On the other hand, gas nuclei will be very significant as nucleation cavities. If inert gas molecules are present in the liquid and, hence, in the bubble, Equations 1 and 5 are modified as foUoWs : 2u -I-
i
1 I
'I
(
P.-P
=
1 ! .
~
7- P a
Equations 6 and 7 show that the superheat required for 1
4.. P L. . . Figure 3. Maximum sypb.'ha;porriblu in a purr liquid. Th &de of Wismsr (24) are indica&d by rhr circbr and are canpared with ihmy applicOblc to a wn der Waals subsiance 42
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
T h o d y n a m i c equ2ibrium illustratrd by capillary rirr in a vskm containing a piwe substame Figure 4.
a given size bubble to grow is decreased by the presence of the gas partial pressure., pp. Nucleolion at katod Surfacer
In boiliig systems, nucleation takes place at or certainly very close to the heated solid-liquid interface. Figure 6 represents two greatly enlarged photographs of a heated surface, Westwater (23), taken at successive time intervals. The first picture shows a black spot which we shall postulate as a s m a l l cavity in the surface. The second picture shows a bubble growing over that spot. It is not inconsistent with this observation to asgume that the bubble originated a t the cavity. GrilKth and Walliis (72) carried out nucleation experiments by immersing in water (at various pressures at and below atmospheric) copper sheets with a variety of cavities. The water was raised in temperature and the superheat measured when bubbles formed repeatedly at a cavity. The superheat calculated by means of the author's Equation 5, using cavity radius, agreed well with their data. Using the evidence just outlined above, the author suggests that a surface contains many cavities, and bubbles form at a heated surface from cavities which already have some gas or vapor presenMo-called active cavities. When heat is added, the vapor pocket in an active cavity grows by evaporation at the liquidvapor interface near the heated wall, as shown in Figure 70. In a liquid at or near the saturation temperature, the bubble grows and detaches, trapping vapor in the cavity as shown in Figure 7b. This trapped vapor is the nucleus for the next bubble. Usually a bubble encompasses many cavities before it detaches; thus, this vapor-trapping process can induce an inactive cavity (one entirely filled with the pure liquid) into activity. When the liquid is highly subcooled, the bubble grows and collapses while attached to the surface. Even in this case, the vapor pocket remains in the cavity, ready for the next bubble to grow. It is not necessary that inert gas be present. Some cavities can contain pure vapor even at very highly subcooled temperatures. Two cases of different contact 0 angle 0 and opposite curvature of the interface may be considered : p. > p in Figure 7c and p. < p in Figure 7d. As the surface is allowed to cool below the saturation temperature of the liquid, the cavity in Figure 7c, if it contained pure. vapor, would collapse and be completely filled with liquid, thus becoming inactive. As cavity d cools down, however, the interface recedes into the cavity, decreasing the radius of curvature and reducing p.. Since p. decreases, TW6of the vapor decreases; the cavity does not collapse and is ready as an active nucleus when the surface is subequently heated. Gri5th and Wallis (72) obtained the curvul in Figurc Ea by counting nucleation ipom n, number per aquare foot, an (T, T,,J was varied on a clean copper surface (finished with 3/0 emery paper) at the bottom of a pool of liquid. Cavity radius r calculated from Equation 5 brings all three of these curves together as shown in Figure 86. This curve is essentially an integration of a
I
---h
1s
Figure 5. Approiimation of vapor-liquid equilibrim at a nawd interface of a vapor bubble in a onc-cmponcnf s y s h
Figure 6. Mqnijedfihtogr@hs by Wesfwaterand Santangelo (23) shouting (a) a ptobable camty in a hated surface and (b) a k m f r m showing s b d b k growirg over the cam0
n
Figure 7.
Theformarion of
bubbbs of vapor over cnuitiar in a hated
swfa
-
AUTHOR Wawm M . RohraMW i s Rofesssor of Mechanical EngiMaing of the Massachusetts &itu& of Tecfwdpgy. VOL 5 8
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20 10
Figure 8. NucIea!~on on 4 heated horiwntal s w f m immsed in various liquids. Crifith and WdIis (12) counted the number of nucleaIion sites (./A) os a funchon of the hgree of suparhror (a)
Figure 9. The tnitiarion of bubbk growth from a fube wall with the Juid in forced cawechon. D4k2 are from Eagles and Rohraow (3)
forced cornchon in thdsystm of Figure 9
cavity distribution function such as that in Figure 8integration being performed from I = rn to Y for the last cavity to be activated and n, being the number of cavities between size range Y and Y &. The curve characterizes ,the particular surface for boiliig heat transfer. Bankoff (1) extended the thermodynamic nucleation theory of Volmex (27) and Fisher (9) to conrider wetted and unwetted surface projections, plane surfaces, and cavities. He concludes that nucleation at truly plane surfaces, like nucleation in pure liquids, is of no importance in surface boiling. I n addition, he demonstrates that a surface projection, thought previously to sewe as a nucleation center, is inferior to a plane surface' in minimizing the(work of forming a vapor nucleus. Then only cavities should be of importance as nucleation sites, beiig effective only if poorly wetted by the liquid or if they contain noncondensable gas. A well wetted cavity initially containing noncondensable gas may lose its gas by diffusion as boiling continues. I n
thii case, the cavity will continue to nucleate from the entrapped vapor but will completely fillwith liquid and become inactive when the surface is allowed to cool. Photographic evidence (7) tends to verify these conclusions. Diethyl Ether and *pentane were boiled on clean, poliihed heating surfaces of zinc and of an aluminum alloy. Pictures were taken through a microscope. In one run 20 nucleation sites were observed-13 were identified as piis in the surface (0.0003 to 0.0033 inch in diameter), three were scratches (about 0.0005 inch wide), three were at boundaries between the heater plate and the sealing cement, and one was a shifting particle of unidentified material which appeared briefly. Microscopic observations of boiling were also made on numerous polycrystalline surfaces and for comparison on a single crystal of zinc and on polycrystalline zinc. No significant difference in the heat fluxus. temperature difference curve was observed and no grain boydaries were seen to be acting as nucleating sites.
+
I
and os a fmtion of cavrty radius, r (6). The dirm'bunon of wtiue cavities accmding to their radii r's shown in (c!
U
INDUSTRIAL AND ENGINEERING CHEMISTRY
Figwe 10. Expnirnental detmimtion of incipiunt baling wtth
Inciplen! Boiling in Forced Convection
Consider a liquid flowing in a tube. As the heat flux or wall temperature is raised, boiling or nucleation begins'at'a particular value of the wall temperature. We wish to investigate this condition. The following procedure was'dekloped by Bergles and Rohsenow (3) based on a suggestion of Hsu and Graham (74). In this flow inside the tube fhe expression'for heat flux is
Here h is the heat transfer coefficientwhich is a function of the geometry, the'fluid properties, and the flow rate; k, is the liquid thermal Conductivky. For a given liquid temperature, both the temperature gradient (d'i"/&~)~:o and. the wall temperature 'i", &crease as the heat flux increases. A seiies'of curves representing the temperature distribution very near the heated wall is shown for incrksing heat flux in Figure 9. Also shown in F i p r e 9 is a curve l a w e d T,*, which is a plot of Fquation 5 with the radius of the cavity plotted-as the distance from the heated 'surface. A posshle theory: Nucleation takes place when the temperatbe curve in the liquid is tangent to t h e ' c ~ erepresenting 'Equation 5. -e impkcation is that.the surface contains cavities of various sizes (Figure 8c) and when the temperature at the outer surface of the bubble reach% the criticd value given by Equation 5, the bubble ROWS at the cavity whose radius hepresented by the distadce W e e n the wall and the poiit of intersection. This nucleation theory was verified by a wide variety of data (3,'Figure 10 representing one set. The ordinate represents the difference between the measured
heat flux and that predicted by extrapolation for nonboiling. . The arrows represent the magnikdes 'of T, calculated for the condition of tangency in Figure 9. Boiling begins where the data points in Figure 10 rise. Excellent ageement is observed. ins!ability in Boiling Liquid Melair
Many experimenters have observed a kind of instability when boiling alkali metals which has not been observed in boiling liquid nonmetals. The effect shown by the data in Fi&& 11 and 12 is represenetivc of that' observed by many investigators: For example, b i h g took place on a horizontal flat disk, 2.5 inches in diameter, at'the bottom of a sodium pool. At a fixed heat flhx the wall tbpeiature rose and, when nucleation began, it fell rathex suddenly, and then rose again. "he lieat f l was ~ maintained steady but the surface tempeiature rose alternately and fell as shown in Figure 11.' The coFespondingmaximum and minimum points are shown .in Figure 12 for a stainless. steel surface. Shown also in Figure 12 are data for a &&el surface which, with the same lap finish, boiled with stability and did not exhibit these excursions in wall temperatures. A sound pick-up attached to the apparatus indicated a significant noise at.the minimum temperature (boiling) and no noise at the maximum' temperature (no boiling). Many surfaces were studied (78). On some, boiling was stable; on others, it was unstable. Mirror-finished nickel surfaces w e e stable; the same surface etched was unstable. S d a c e s with manufactured small cylindrical cavities were unstable; those with reentrant cavities were stable. In each case the instability disappeared beyond some upper limit of heat flux. It is suggested (78) that under the conditions at which this unstable 'boiling can -exist, the active nucleating
Figure 77. Tnnpnphachisw of apoint on n 376-stinlcsssfcddisk Figure 72. T h gect of d@rtnt marninlr on the stability of nucleute hating a. pool of molfm sodium, as m a y e d by Mmlo and R O ~ C M W boiling of mol@ dim. Cbrad nrcks rafer to 376-stainless stet, (78), fm wrious hatJlurcs hating stuface, open nrcks rifer to "A" n i c M stufaces. D a t a poino with CTOSS(I ¬e non8mling data unth the dashed line connecting p%nts t a b %fora bump" and "of& btbnp" of the smnc power lecd V O L 5 8 NO. 1 J A N U A R Y 1 9 6 6
45
1101111
Figwe 73. Diagrurn of propored nucleonon cyclc showing interjam pm'tion and liquid ranpnoture dishibvrion
cavities can become inactive by the liquid's rushing in behind a departing bubble. In those cases it is visualized that the liquid can reach the bottom of the cavity, filling the cavity and deactivating it. After bubble departure, we mume that cold liquid at a temperature To rushes down into the cavity mouth (F13). At the same time, it is assumed the surrounding wall of the cavity has cooled to a minimum temperature identical to this cold liquid temperature, To. The author suggests that TOis slightly larger than the saturation temperature at the liquid pressure, but less than the saturation temperature at the p m u r e of the vapor within the cavity, and that the size depends on the heat fluxper unit area, the wall material, and bubble frequency. Due to the acceleration of the liquid, the inertia forces may be large. However, after traveling a very short distance into the cavity, because of the small dimensions involved, it is asaumed these inertia force are damped out and can be neglected. Because of the curvature of the liquid-vapor interface, the vapor trapped inside the cavity is at a higher pressure than the liquid pressure. Because To is less than the saturation temperature of this vapor, condensation begins to occur. The interface recedes into the cavity, liiited by the rate at which the latent heat of vaporization, supplied at the minkus, can be conducted away by the liquid. As the l4uid travds into the cavity it receives heat not only from the condensation of va'por but also by conduction from the cavity walls. The interface continues to slow down as the temperature of the Iiquid increases, and at some point when the bulk liquid temperature wuals the vapor temperature, condensation will stop, the bubble will begin to grow again, and the cavity will remain active. This nucleation cycle is shown schematically in Figure 13. If, on the other hand, the bulk liquid temperature never reaches the vapor temperature, all the vapor will continue to condense. l e a d m to complete collause of the cavity. The above mechanism is very approximate, containing many inherent assumptions. However, it does establish a physical model which can be analyzed in an attempt to explain nucleation stability. A number of investigators (5, 77, 79) have shown that the wall temperature under a bubble during nucleation, growth, and departure is similar to that sketched in Figure 14. The decreasing wall temperature corresponds to a significant increase in the local heat transfer from the surface when the bubble forms and grows while attached to the surface. Using this information, Marto (78) solved the a p propriate heat conduction equation for a moving surface to determine penetration history as shown in Figure 13. His result for penetration distance x us. time has the following form:
-
x =
Afi-
Bf
(9)
~i~~~~74. &ha r8prermt& fic .f & iota[ nUf.c6 botjw where A and B are collections of quantities involving a d h a 1 wfmfanpnahaa during bubblegrowthfrom a cam0 fluid properties, cavity geometry, and heat flux. Setting 46
INDUSTRIAL AND ENGINEERING CHEMISTRY
dx/dt = 0, the maximum penetration depth x * is found
to be
According to this analysis, then, for a cavity to remain active, 7 ” = x / L , where L is the cavity length, should be less than unity. The following is the result obtained (78) for 7 ” :
(
)
$ - ~ ~ ~ ~ 1 ’ ~ ~ ~
1 ,Lc( 1W+ sin e)
(11)
Actually, since there are a number of simplifying assumptions in the analysis, the criterion of 17” = 1.0, dividing stability and instability should not be considered as a precise magnitude. Rather, the magnitude of v* should be used as a relative quantity-the greater v*, the more chance that unstable boiling will exist. The above prediction suggests that as heat flux is raised, v* decreases; hence, the observed upper limit of heat flux for the unstable operation range is to be expected. Also, if the liquid does not wet the solid (6 > 90°), boiling should be stable. The following are magnitudes of v* calculated for a few different liquids boiling on stainless steel at a heat flux of 50,000 B.t.u./hr. per sq. ft. with an assumed cavity size of 0.0024-inch diameter and 0.012-inch length. Liquid Methyl alcohol Liq Nz HzO HzO
HzO
Contact Angle 46 O0 60 O 80’ O0
Na
O0
4*
0.0000137 0.000033 0.0001 54 O.OO0017 0.00116 6.23
These magnitudes show that sodium may indeed experience unstable boiling while the nonmetals should be quite stable. In Figure 12 the stainless steel surface was unstable while the nickel surface was stable. For the same assumed cavity size
*
rlss- 1.9 TNi*
suggesting that the observed result could be expected. Conclusion
For nucleation at heating surfaces it appears that the theories of homogeneous nucleation are not applicable. Instead, a theory of nucleation at cavities on the surface is more plausible and is consistent with experimental observations. NOMENCLATURE a
b
C
empirical exponent from ref. (73)which varies with surface roughness = empirical coefficient defined by 2 Aq/Qo, and approximately equal to 20 = empirical coefficient from ref. ( 7 3 ) which varies with surface roughness =
CL C, h hf, J
specific heat of the liquid specific heat of the boiler wall heat transfer coefficient latent heat of vaporization = mechanical equivalent of heat kL = thermal conductivity of the liquid k, = thermal conductivity of the boiler wall L = overall depth of a cavity p = pressure in the liquid pv = pressure in the vapor pg = partial pressure of noncondensable gas q o = q / A = heat flux per unit area in the boiler wall Aq = step change in heat flux near a cavity during bubble growth r = radius of hemispherical bubble on a surface R = radius of curvature of liquid-vapor interface R, = cavity radius R, = gas constant t = time during interface penetration into a cavity T = temperature T L = temperature of the liquid T , = T , = temgerature of the liquid-vapor interface T O = initial temperature of both the liquid and the boiler wall during interface penetration into a ca>ity T,,, = local maximum surface temperature of the boiler wall during bubble nucleation at a cavity T,,, = local minimum surface temperature of the boiler wall during bubble nucleation at a cavity Tsst = saturation temperature at the liquid pressure T, = surface temperature of the boiler wall T , = T , = saturation temperature at the vapor pressure x = liquid-vapor interface penetration distance into a cavity during bubble nucleation x* = maximum interface penetration distance into a cavity y = distance into the liquid, measured from the boiler surface u = specific volume CXL = thermal diffusivity of the liquid cyzD = thermal diffusivity of the boiler wall E = empirical coefficient, less than one, used to approximate the wall temperature rise during the bubble waiting period = x*/L = dimensionless interface penetration distance ?* u = surface tension = density of the liquid p~ = density of the vapor pv = density of the boiler wall = dynamic contact angle = = = =
?
REFERENCES (1) Bankoff, S. G., J. Heat Transfer 79, 735 (1957). (2) Becker, R., Doring, W., Am. Physik 24, 71 9 (1 935). (3) Bergles, A. E., Rohsenow, W. M., J. Heal Transfer 86, 365 (1964). (4) Bernath, L., IND.ENG.CHEM.44, 1310 (1952). (5) Bonnet, C., Macke, E., Morin, R., “Visualisation de L’ebullition Nuclee de L’eau iPression Atmospherique et Mesure SimultanCe des Variations de Temperature de Surface,” E U R 1622.f, Ispra, Italy (1964). (6) Briggs, L. J., J. Appl. Phys. 21, 721 (1950). (7) Clark H B Strenge, P. S., Westwater, J. W., Chem. Eng. Prog. Symp. Ser. 55 id3 ( i g i 4 j . (8) Clark, J. A,, Tech. Rept. No. 7, Heat Transfer Laboratory, M I T (1956). ( 9 ) Fisher, J. C., J.Appl. Ph,ys. 19, 1062 (1948). (10) Frenkel, J., “Kinetic Theory of Liquids,” Oxford Univ. Press, Oxford, 1948. (11) Gibbs, J. W., “Collected Works,” Vol. I, Yale Unirersity Press, New Haven, 1948. (12) Griffith, P., Wallis, J. D., A.I.Ch.E. Symp. Ser., No. 31 (1960). (13) Hsu, S. T., Schmidt, F. W. “Measured Variations in Local Surface Temperatures in Pool Boiling of Watbr,’) ASME-AIChE Heat Transfer Conf., Buffalo, 1960.
w,
(14) Hsu, Y . Y . , Graham, R. W., N A S A Tech. Nofa T N P-594 (May 1961). (15) Keenan, J . H., “Thermodynamics,” Wiley, New York, 1942. (16) LePpert, G . , C.C., “Advances Heat Transfer,” Academic Press, 1964. (17) Marcus, B. D., Ph.D. Thesis, Cornel1 University, 1963. (18) Marto, P. J., Rohsenow, W. M., Rept. 5219-33, Heat Tlansfer Laboratory, M I T (1965). (19) Moore, F. D.,Mesler, R., Am. Insf. Chem. Eng.J. 7 (4), pp. 620-4 (1961). (20) Rohsenow, W. M., “Developments in Heat Transfer,” M I T Press, Cambridge, 1964. (21) Volmer, M., “Kinetik der Phasenbildung,” Steinkopf, Dresden and ~ ~ (1939). (22) Westwater, J. W., “Advan. Chem. Eng.,” Academic Press, 1956. (23) Westwater, .I. W., Santangelo, J. G., IND.END.CHEM.47, 1605 (1955). (24) Wismer, K. L., J. Phys. Chem. 26, 301 (1922).
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